Results 1 
7 of
7
Hypercomputation and the Physical ChurchTuring Thesis
, 2003
"... A version of the ChurchTuring Thesis states that every e#ectively realizable physical system can be defined by Turing Machines (`Thesis P'); in this formulation the Thesis appears an empirical, more than a logicomathematical, proposition. We review the main approaches to computation beyond Turing ..."
Abstract

Cited by 21 (0 self)
 Add to MetaCart
A version of the ChurchTuring Thesis states that every e#ectively realizable physical system can be defined by Turing Machines (`Thesis P'); in this formulation the Thesis appears an empirical, more than a logicomathematical, proposition. We review the main approaches to computation beyond Turing definability (`hypercomputation'): supertask, nonwellfounded, analog, quantum, and retrocausal computation. These models depend on infinite computation, explicitly or implicitly, and appear physically implausible; moreover, even if infinite computation were realizable, the Halting Problem would not be a#ected. Therefore, Thesis P is not essentially di#erent from the standard ChurchTuring Thesis.
AXIOMATIZING MATHEMATICAL CONCEPTUALISM IN THIRD ORDER ARITHMETIC
"... Abstract. We review the philosophical framework of mathematical conceptualism as an alternative to settheoretic foundations and show how mainstream mathematics can be developed on this basis. The paper includes an explicit axiomatization of the basic principles of conceptualism in a formal system C ..."
Abstract
 Add to MetaCart
Abstract. We review the philosophical framework of mathematical conceptualism as an alternative to settheoretic foundations and show how mainstream mathematics can be developed on this basis. The paper includes an explicit axiomatization of the basic principles of conceptualism in a formal system CM set in the language of third order arithmetic. This paper is part of a project whose goal is to make a case that mathematics should be disassociated from set theory. The reasons for wanting to do this, which I discuss in greater detail elsewhere ([22]; see also [19] and [23]), involve both the philosophical unsoundness of set theory and its practical irrelevance to mainstream mathematics. Set theory is based on the reification of a collection as a separate object, an elementary philosophical error. Not only is this error obvious, it also has the spectacular consequence of immediately giving rise to the classical set theoretic paradoxes. Of course, these paradoxes are not derivable in the standard axiomatizations of set theory, but that is only because these systems were specifically designed to avoid them. In these systems the paradoxes are blocked by means of ad hoc restrictions on the set concept that have no obvious intuitive justification, which has led to the development of a large literature of attempted rationalizations
Forthcoming in Minds and Machines, 2011. On the Possibilities of Hypercomputing Supertasks 1
, 2010
"... This paper investigates the view that digital hypercomputing is a good reason for rejection or reinterpretation of the ChurchTuring thesis. After suggestion that such reinterpretation is historically problematic and often involves attack on a straw man (the ‘maximality thesis’), it discusses prop ..."
Abstract
 Add to MetaCart
This paper investigates the view that digital hypercomputing is a good reason for rejection or reinterpretation of the ChurchTuring thesis. After suggestion that such reinterpretation is historically problematic and often involves attack on a straw man (the ‘maximality thesis’), it discusses proposals for digital hypercomputing with “Zenomachines”, i.e. computing machines that compute an infinite number of computing steps in finite time, thus performing supertasks. It argues that effective computing with Zenomachines falls into a dilemma: either they are specified such that they do not have output states, or they are specified such that they do have output states, but involve contradiction. Repairs though noneffective methods or special rules for semidecidable problems are sought, but not found. The paper concludes that hypercomputing supertasks are impossible in the actual world and thus no reason for rejection of the ChurchTuring thesis in its traditional interpretation. 1
Transfinite Machine Models
, 2011
"... In recent years there has emerged the study of discrete computational models which are allowed to act transfinitely. By ‘discrete ’ we mean that the machine models considered are not analogue machines, but compute by means of distinct stages or in units of time. The paradigm of such models is, of co ..."
Abstract
 Add to MetaCart
In recent years there has emerged the study of discrete computational models which are allowed to act transfinitely. By ‘discrete ’ we mean that the machine models considered are not analogue machines, but compute by means of distinct stages or in units of time. The paradigm of such models is, of course, Turing’s original
A New Problem for Rule Following
"... This is part of an extended argument of mine about the ChurchTuring thesis (CTT). In Hogarth 1994 I argued that the thesis is a thoroughly empirical claim. In Hogarth 2004, 2008 I rejected that view, arguing instead that the thesis is really a pseudoproposition like ‘Australia is below England’, or ..."
Abstract
 Add to MetaCart
This is part of an extended argument of mine about the ChurchTuring thesis (CTT). In Hogarth 1994 I argued that the thesis is a thoroughly empirical claim. In Hogarth 2004, 2008 I rejected that view, arguing instead that the thesis is really a pseudoproposition like ‘Australia is below England’, or, better, like ‘Euclidean geometry is
Report # CSR 102005Newtonian systems, bounded in space, time, mass
"... Newtonian systems, bounded in space, time, mass and energy can compute all functions by ..."
Abstract
 Add to MetaCart
Newtonian systems, bounded in space, time, mass and energy can compute all functions by